On the Hodograph Method in Magnetogasdynamics

Abstract

R Yur'ev 1 discussed the Chaplygin-type equations in magnetogasdynamics. Mathematically speaking, the hodograph method tacitly has been introduced in magnetogasdynamics. The object of this paper is to obtain such an equation in the form of a function ^ representing the streamlines for the plane motion of an electrically conducting, compressible, steady flow with infinite conductivity in the presence of a magnetic field, co-linear with linear momentum of the flow. I t is shown that the flow is essentially similar to the ordinary perfect flow in gasdynamics with an appropriate pressure-density relation. Taniuti has independently found the basic equation for plane flows without making the assumption of small perturbations, showing that a discontinuity surface can occur in slow subsonic flow if the magnetic field is of adequate strength. Taniuti's equation is extended to the general case of threedimensional flow by Imai. Moreover, in Refs. 3 and 4 the hodograph method has not been explicitly introduced. In communications between the present author and Prof. W. R. Sears, there have been more explicit introductions of the hodograph method in magnetogasdynamics, but these are as yet unpublished. However, the work in this paper was developed by the author independently at the same time.